Torsion of elliptic curves and unlikely intersections

TL;DR

This paper investigates the unlikely intersections involving torsion points on elliptic curves and their images on the projective line, providing effective bounds and insights into their distribution.

Contribution

It introduces effective methods to analyze unlikely intersections of torsion points on elliptic curves with their images on the projective line.

Findings

Derived explicit bounds for intersections

Identified conditions for finiteness of intersections

Enhanced understanding of torsion point distributions

Abstract

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.

Equations

$$\pi :E\to {\mathbb{P}}^1,$$

$${\pi }_1(E_1[\infty ])\cap {\pi }_2(E_2[\infty ])$$

$${\pi }_1(E_1[\infty ])\cap {\pi }_2(E_2[\infty ])<c.$$

$$S=\{s_1,\dots ,s_n\},S^{\prime }:=\{s_1^{\prime },\dots ,s_n^{\prime }\}\subset {\mathbb{P}}^1(\bar{k})$$

$$\begin{array}[]{rcc}E&\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}&E\\ x&\mapsto&-x\end{array}$$

$$\pi :E\to E/\iota ={\mathbb{P}}^1$$

$$r:=\{r_1,r_2,r_3,r_4\}\subset {\mathbb{P}}^1(\bar{k}),$$

$${\pi }_r:E_r\to {\mathbb{P}}^1$$

$${\pi }_r(E_r[3])\sim \{1,{\zeta }_3,{\zeta }_3^2,\infty \},$$

$${\pi }_r(E_r[4])\sim \{0,1,-1,i,-i,\infty \}.$$

$${\pi }_r(E_r[n])\cap {\pi }_{r^{\prime }}(E_{r^{\prime }}[n^{\prime }]),n,n^{\prime }\in \mathbb{N},$$

$$r\subset {\pi }_{r^{\prime }}(E_{r^{\prime }}[\infty ])?$$

$$r\subset \tilde{L}\Rightarrow {\pi }_r(E_r[\infty ])\subseteq \tilde{L}?$$

$$z\mapsto -z\text{ and }z\mapsto 1/z,$$

$$P_{\lambda }\mapsto {\mathrm{Sym}}^n(P_{\lambda })$$

$$j_n:{\mathcal{P}}_n={\mathrm{Sym}}^n(\mathcal{P})\to {\mathbb{P}}^1.$$

$${\mathrm{Sym}}^n(P_{\lambda })\to {\mathcal{M}}_{0,n}(P_{\lambda })\simeq {\mathcal{M}}_{0,n},$$

$${\mu }_n:{\mathcal{P}}_n\to {\overline{\mathcal{M}}}_{0,n}$$

$$X(N)\to Y(N):=X(N)/\iota .$$

$$Y:={\cup }_{N\in \mathbb{N}}Y(N)$$

$${\mathrm{Sym}}^n(Y)\hookrightarrow {\mathcal{P}}_n\to {\overline{\mathcal{M}}}_{0,n}.$$

$${\mu }_4:Y_{4,\omega }\to {\overline{\mathcal{M}}}_{0,4}={\mathbb{P}}^1$$

$$({\mu }_4,j):Y_{4,\omega }\to {\overline{\mathcal{M}}}_{0,4}\times {\mathbb{P}}^1$$

$${\mu }_5:Y_{5,\omega }\to {\overline{\mathcal{M}}}_{0,5}$$

$${\mu }_6:Y_{6,\omega }\to {\overline{\mathcal{M}}}_{0,6}$$

$$(0,\infty ),(1,-1),(i,-i)$$

$$({\mathbb{P}}^1{)}^4\stackrel{cr}{⟶}(\mathbb{Z}/2\oplus \mathbb{Z}/2)\backslash ({\mathbb{P}}^1{)}^4/{\mathrm{PGL}}_2={\mathbb{P}}_1^1\to {\mathfrak{S}}_3\backslash {\mathbb{P}}_1^1.$$

$${\mu }_4:Y_{4,\omega }\to {\mathbb{P}}^1$$

$$z_1,z_2\in W_n/\iota \text{ and }{z}_3,z_4\notin W_n/\iota .$$

$$(z_1-z_3)(z_2-z_4)=(z_2-z_3)(z_1-z_4),$$

Full text

To Nigel Hitchin, with admiration.

Introduction

Let $k$ be a field of characteristic $\neq 2$ and $\bar{k}/k$ an algebraic closure of $k$. Let $E$ be an elliptic curve over $k$, presented as a double cover

[TABLE]

ramified in 4 points, and $E[\infty]\subset E(\bar{k})$ the set of its torsion points. In [1] we proved:

Theorem 1**.**

If $E_{1},E_{2}$ are nonisomorphic elliptic curves over $\bar{{\mathbb{Q}}}$, then

[TABLE]

is finite.

Here, we explore effective versions of this theorem, specifically, the size and structure of such intersections (see [5] for an extensive study of related problems). We expect the following universal bound:

Conjecture 2** (Effective Finiteness–EFC-I).**

There exists a constant $c>0$ such that for every pair of nonisomorphic elliptic curves $E_{1},E_{2}$ over ${\mathbb{C}}$ we have

[TABLE]

We say that two subsets of the projective line

[TABLE]

are projectively equivalent, and write $S\sim S^{\prime}$, if there is a $\gamma\in{\rm PGL}_{2}(\bar{k})$ such that (modulo permutation of the indices) $s_{i}=\gamma(s_{i}^{\prime})$, for all $i$.

Let $E$ be an elliptic curve over $k$, $e\in E$ the identity, and

[TABLE]

the standard involution. The corresponding quotient map

[TABLE]

is ramified in the image of the 2-torsion points of $E(\bar{k})$. Conversely, for

[TABLE]

the double cover

[TABLE]

with ramification in $r$ defines an elliptic curve; given another such $r^{\prime}$, the curves $E_{r}$ and $E_{r^{\prime}}$ are isomorphic (over $\bar{k})$ if and only if $r\sim r^{\prime}$, in particular, the image of 2-torsion determines the elliptic curve, up to isomorphism.

Let $E_{r}[n]\subset E_{r}(\bar{k})$ be the set of elements of order exactly $n$, for $n\in{\mathbb{N}}$. The behavior of torsion points of other small orders is also simple:

[TABLE]

where $\zeta_{3}$ is a nontrivial third root of 1, and

[TABLE]

In particular, up to projective equivalence, these are independent of $E_{r}$. However, for all $n\geq 5$, the sets $\pi_{r}(E_{r}[n])$, modulo ${\rm PGL}_{2}(\bar{k})$, do depend on $E_{r}$, and it is tempting to inquire into the nature of this dependence.

In this note, we study $\pi_{r}(E_{r}[n])$, for varying curves $E_{r}$ and varying $n$. Our goal is to establish effective and uniform finiteness results for intersections

[TABLE]

for elliptic curves $E_{r},E_{r^{\prime}}$, defined over $k$. We formulate several conjectures in this direction and provide evidence for them.

The next step is to ask: given elliptic curves $E_{r},E_{r^{\prime}}$ over $\bar{k}$, when is

[TABLE]

We modify this question as follows: Which minimal subsets $\tilde{L}\subset{\mathbb{P}}^{1}(\bar{k})$ have the property

[TABLE]

The sets $\tilde{L}$ carry involutions, obtained from the translation action of the 2-torsion points of $E$ on $E$, which descends, via $\pi$, to an action on ${\mathbb{P}}^{1}$ and defines an embedding of ${\mathbb{Z}}/2\oplus{\mathbb{Z}}/2\hookrightarrow{\rm PGL}_{2}(\bar{k})$. It is conjugated to the standard embedding of ${\mathbb{Z}}/2\oplus{\mathbb{Z}}/2$, generated by involutions

[TABLE]

acting on $\tilde{L}$. This observation is crucial for the discussion in Section 4, where we prove that, modulo projectivities, $L:=\tilde{L}\setminus\{\infty\}$ are fields.

Acknowledgments: The first author was partially supported by the Russian Academic Excellence Project ‘5-100’ and by Simons Fellowship and by EPSRC programme grant EP/M024830. The second author was supported by the MacCracken Program offered by New York University. The third author was partially supported by NSF grant 1601912.

1. Generalities

Let $j:\mathcal{E}\rightarrow{\mathbb{P}}^{1}$ be the standard universal elliptic curve, with $j$ the $j$-invariant morphism. Consider the diagram

\textstyle{E_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\subset}$$\textstyle{P_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\subset}$$\textstyle{\mathcal{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\textstyle{\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{{\mathbb{P}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{1}}

assigning to each fiber $E_{\lambda}:=j^{-1}(\lambda)$ the quotient $P_{\lambda}=\pi(E_{\lambda})\simeq{\mathbb{P}}^{1}$, by the involution $\iota:x\mapsto-x$ on $E_{\lambda}$. (This is well-defined even for singular fibers of $j$.)

Note that $\mathcal{P}\rightarrow{\mathbb{P}}^{1}$ is a ${\rm PGL}_{2}$-torsor. Taking fiberwise $n$-symmetric product:

[TABLE]

we have associated ${\rm PGL}_{2}$-torsors

[TABLE]

Taking ${\rm PGL}_{2}$-invariants, we have a canonical projection

[TABLE]

to the moduli space of $n$-points on ${\mathbb{P}}^{1}$. The associated ${\rm PGL}_{2}$-torsor is trivial; fixing a trivialization we obtain a morphism

[TABLE]

For every $N\in{\mathbb{N}}$, we have the modular curve $X(N)\rightarrow{\mathbb{P}}^{1}$, parametrizing pairs of elliptic curves together with $N$-torsion subgroups. The involution $\iota$ induces an involution on every $X(N)$, we have the induced quotient

[TABLE]

Since the family $j:\mathcal{E}\rightarrow{\mathbb{P}}^{1}$ has maximal monodromy ${\rm SL}_{2}({\mathbb{Z}})$, the curves $X(N)$ and $Y(N)$ are irreducible. We have a natural embedding $Y(N)\hookrightarrow\mathcal{P}$. Put

[TABLE]

and consider

[TABLE]

Note that ${\rm Sym}^{n}(Y)$ is a union of infinitely many irreducible curves, each corresponding to an orbit of the action of the monodromy group ${\rm PGL}_{2}({\mathbb{Z}})$ on the generic fiber of the restriction of $j_{n}$ to ${\rm Sym}^{n}(Y)$. Let $Y_{n,\omega}\subset{\rm Sym}^{n}(Y)$ be an irreducible component corresponding to a ${\rm PGL}_{2}({\mathbb{Z}})$-orbit $\omega$ (for the monodromy action, as above). We now formulate conjectures about $\mu_{n}$, for small $n$, which guide our approach to the study of images of torsion points.

Conjecture 3**.**

The map

[TABLE]

is finite surjective, for all but finitely many $\omega$.

Conjecture 4**.**

The map

[TABLE]

is a rational embedding, for all but finitely many $\omega$.

Conjecture 5**.**

The map

[TABLE]

is a rational embedding, for all but finitely many $\omega$. Moreover, if for some distinct orbits $\omega$ and $\omega^{\prime}$ the corresponding images $\mu_{5}(Y_{5,\omega})$ and $\mu_{5}(Y_{5,\omega^{\prime}})$ are curves, then they are different.

Conjecture 6**.**

The map

[TABLE]

is a rational embedding, for all but finitely many $\omega$. Moreover, if $\mu_{6}(Y_{6,\omega})$ is a curve then there exist at most finitely many $\omega^{\prime}$ such that

• •

$\mu_{6}(Y_{6,\omega^{\prime}})$* is a curve and*

• •

$\mu_{6}(Y_{6,\omega})\cap\mu_{6}(Y_{6,\omega^{\prime}})\neq\emptyset$.

2. Examples and evidence

We now discuss examples and evidence for Conjectures in Section 1.

Example 7*.*

We have

• •

$\mu_{4}({\rm Sym}^{4}(Y(2))\simeq\overline{\mathcal{M}}_{0,4}={\mathbb{P}}^{1}$,

• •

$\mu_{4}({\rm Sym}^{4}(Y(3))$ is a point in $\overline{\mathcal{M}}_{0,4}$.

Consider ${\rm Sym}^{4}(Y(4))$. Note that $\pi(E[4])=\{0,1,-1,i,-i,\infty\}$ is an orbit of the symmetric group $\mathfrak{S}_{4}$, acting on ${\mathbb{P}}^{1}$. The pairs

[TABLE]

are pairs of stable points for $3$ even involutions in $\mathfrak{S}_{4}$, and the action of $\mathfrak{S}_{4}$ is transitive on pairs and inside each pair. There are two different $\mathfrak{S}_{4}$-orbits of $4$-tuples: either the orbit contains two pairs of vertices such as $(0,\infty),(1,-1)$, or a pair and two points from different pairs $(0,\infty),(1,i)$. Thus ${\rm Sym}^{4}(Y(4))$ has two components which project to different points modulo ${\rm PGL}_{2}$; therefore, there exist exceptional orbits $\omega$ such that $\mu_{4}(Y_{4,\omega})$ is a point.

Lemma 8**.**

If $\mu_{4}(Y_{4,\omega})$ is a point then all cross ratios of 4-tuples of points parametrized by $Y_{4,\omega}$ are constant.

Proof.

The map $\mu_{4}$ can be viewed as a composition

[TABLE]

Thus we have a diagram

\textstyle{({\mathbb{P}}^{1})^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cr\quad\quad\quad}$$\textstyle{({\mathbb{Z}}/2\oplus{\mathbb{Z}}/2)\backslash({\mathbb{P}}^{1})^{4}/{\rm PGL}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{S}_{3}}$$\textstyle{\mathfrak{S}_{4}\backslash({\mathbb{P}}^{1})^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{4}\backslash({\mathbb{P}}^{1})^{4}/{\rm PGL}_{2}}

Note that any irreducible $Y_{4,\omega}$ lifts to a union of connected components $Y_{4,\omega,i}\subset({\mathbb{Z}}/2\oplus{\mathbb{Z}}/2)\backslash Y^{4}$, where cross-ratio is well defined. Thus if $\mu_{4}$ is a rational function of cross-ratio on any four-tuple of points and if $\mu_{4}$ is constant then the cross-ratio is also constant. ∎

Proposition 9**.**

There exist orbits $\omega$ such that

[TABLE]

is surjective.

Proof.

The singular fiber $\mathcal{E}_{\infty}:=j^{-1}(\infty)$ is an irreducible rational curve with one node $p_{\infty}$. The group scheme $\cup_{d\mid n}\mathcal{E}[d]$, whose generic fiber is isomorphic to ${\mathbb{Z}}/n\oplus{\mathbb{Z}}/n$, specializes to $\{\zeta_{n}^{i}\}\subset\mathbb{G}_{m}=\mathcal{E}_{\infty}\setminus p_{\infty}$. Let $\mathcal{E}_{\infty}[n]$ be the specialization of $\mathcal{E}[n]$; then

• •

$\mathcal{E}_{\infty}[n]\subset\{\zeta_{n}^{i}\}$,

• •

there exists a subgroup scheme $\mathcal{W}_{n}\simeq{\mathbb{Z}}/n\subset{\mathbb{Z}}/n\oplus{\mathbb{Z}}/n$ in the group scheme of points killed by $n$, specializing to $\mathcal{E}_{\infty}$, while the complemenary branches specialize to $p_{\infty}$.

Taking the quotient by $\iota$, we find that $(({\mathbb{Z}}/n\oplus{\mathbb{Z}}/n)\setminus{\mathbb{Z}}/n)/\iota$ specializes to [math] in the fiber ${\mathbb{P}}^{1}_{\infty}$ and all other points specialize to subset in $({\mathbb{Z}}/n)/\iota$; the limit depends on the selected direction of specialization.

Assume that we have distinct points $\{z_{1},z_{2},z_{3},z_{4}\}\subset\pi(E[n])$, for a smooth fiber $E$ of $\mathcal{E}$, such that

[TABLE]

The $z_{1},z_{2}$ can be specialized to different nonzero points in $\mathcal{E}_{\infty}/\iota$, and $z_{3},z_{4}$ will specialize to [math].

Assume that $\mu_{4}$ is constant, i.e., the cross-ratio is constant. Since $z_{3},z_{4}$ will specialize to [math], the cross-ratio equals 1. Then

[TABLE]

and

[TABLE]

Near the special fiber, $z_{3}\neq z_{4}$, thus $z_{1}=z_{2}$, contradiction. Thus on orbits of this type, $\mu_{4}$ is not constant, hence surjective. ∎

3. Geometric approach to effective finiteness

Let $E:=E_{r},E^{\prime}:=E_{r^{\prime}}$ be elliptic curves. Consider the diagram

\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\times E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}}

where $C\subset E\times E^{\prime}$ be the fiberwise product over the diagonal $\Delta\subset{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. If $r\neq r^{\prime}$ then $C$ has genus $\geq 2$. By Raynaud’s theorem [4],

[TABLE]

is finite, since it is the preimage of $\pi(E[\infty])\cap\pi(E^{\prime}[\infty])\subset\Delta$, the latter set is also finite. This finiteness argument appeared in [1].

Consider the curves $C$ occurring in this construction. We have a diagram

\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\prime}}$$\scriptstyle{\sigma}$$\textstyle{E}$$\textstyle{E^{\prime}}

where $\sigma,\sigma^{\prime}$ are involutions with fixed points $c_{1},c_{2}$ and $c_{1}^{\prime},c_{2}^{\prime}$, respectively. Assume that

[TABLE]

Then the product involution $\sigma\sigma^{\prime}$ on $C\subset E\times E^{\prime}$ has fixed points in the 6 preimages of the points $\{0,1,\infty\}\subset\Delta_{{\mathbb{P}}^{1}}\subset{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (diagonally), i.e., is the hyperelliptic involution. Thus there is an action of ${\mathbb{Z}}/2\oplus{\mathbb{Z}}/2$ on $C$, induced by the covering maps $\pi$ and $\pi^{\prime}$. The curve $C\subset A=E\times E^{\prime}$ has self-intersection $C^{2}=8$ since it is a double cover of both $E$ and $E^{\prime}$ and its class is equal to $2(E+E^{\prime})$.

• •

If the genus $\mathsf{g}(C)=2$ (three such points) then the image of $C$ in its Jacobian $J(C)$ has self-intersection 2. Consider the map

[TABLE]

and let $n$ be its degree. The preimage $\nu^{-1}(C)\subset J(C)$ has self-intersection $8n$. On the other hand, its homology class is equal to $n$ translations of $C$, hence has self-intersection $2n^{2}$, thus $n=4$. Moreover, $\ker(\nu)={\mathbb{Z}}/2\oplus{\mathbb{Z}}/2$, generated by the pairwise differences of preimages of points $\{0,1,\infty\}$. Thus, $J(C)$ is 4-isogenous to $A:=E\times E^{\prime}$ and $\nu(C)$ is singular, with nodes exactly at the preimages of $\{0,1,\infty\}\subset\Delta_{{\mathbb{P}}^{1}}$. Consider a point $c\in C\subset J(C)$ and assume that $\nu(c)$ has order $m$ with respect to $0\in A$. Then $c$ has order $m$ or $2m$ in $J(C)$, with respect to $0\in J(C)$. Hence the corresponding curve $Y(m)\subset{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (viewed as a moduli space of pairs $E,E^{\prime}$) is given as an intersection of genus $2$ curves containing a point of order $m$ or $2m$, respectively. This is a locus in the moduli space $\mathcal{M}_{2}$ of genus 2 curves.

• •

If $\mathsf{g}(C)=3$ (two such points) then there are three quotients of $C$ which are elliptic curves $E_{1},E_{2},E_{3}$, with involutions $\sigma_{i}\in{\mathbb{Z}}/2\oplus{\mathbb{Z}}/2$ fixing $4$ points on $E_{i}$ which are invariant under the hyperelliptic involution given by complement to $\sigma_{j}$. The kernel of

[TABLE]

contains $E_{i}$, for $i,j,k\in\{1,2,3\}$.

• •

If $\mathsf{g}(C)=4$ then $C$ is $C/\sigma_{i}=E_{i},i=1,2$ and $C/\sigma_{1}\sigma_{2}=C^{\prime}$ where $\mathsf{g}(C^{\prime})=2$ and there are exactly two ramification points on $C^{\prime}$.

• •

If $\mathsf{g}(C)=5$ then $C/\sigma_{1}\sigma_{2}=C^{\prime}$ is a hyperelliptic curve of genus $3$ and the covering is an unramified double cover.

Remark 10*.*

Assume that there is $b\in{\mathbb{P}}^{1}$ and a subset $S\subset C(\bar{k})$ such that $S+b\subset C\subset E\times E^{\prime}$. Then

[TABLE]

hence we have at most $8$ points $c_{i}\in{\mathbb{P}}^{1}$ such that for $x$-coordinates $c_{i}+_{1}b=c_{i}+_{2}b$, where the summation $+_{1}$ corresponds to the summation on the first curve and $+_{2}$ on the second.

Remark 11*.*

The construction can be extended to products of more than two elliptic curves. We may consider

[TABLE]

The ramification divisor of $\pi:\mathcal{A}\rightarrow\mathcal{P}$ is a union of products of projective lines. Let $\Delta={\mathbb{P}}^{1}\subset\mathcal{P}$ be the diagonal, there exists canonical identifications $\delta_{i}:{\mathbb{P}}_{i}^{1}\simeq\Delta$. If $p\in\Delta$ is contained in $\delta_{i}(\pi_{i}(E_{i}[\infty]))$, for all $i$, then the preimage of $p$ in $\mathcal{A}$ is contained in the preimage of the diagonal. This is a curve of genus at least 2, provided there exist $E_{i},E_{j}$ with $r_{i}\neq r_{j}$. Then the set of such $p$ is finite. In particular, if $E$ is defined over a number field $k$ and $p$ is defined over a proper subfield, then $p$ is also in the image torsion points of $\gamma(E)$, where $\gamma$ is a Galois conjugation. Hence, the existence of torsion points with $x$-coordinate in a smaller field has a geometric implication.

We expect the following version of Conjecture 2:

Conjecture 12** (Effective Finiteness–EFC-II).**

There exists a constant $c>0$ such that for every elliptic curve $E_{r}$ over a number field and every $\gamma\in{\rm PGL}_{2}(\bar{{\mathbb{Q}}})$ with $\gamma(r)\neq r$ we have

[TABLE]

4. Fields generated by elliptic division

In this section, we explore properties of subsets of ${\mathbb{P}}^{1}(\bar{k})$ generated by images of torsion points, following closely [1]. For

[TABLE]

a set of four distinct points, let $E_{r}$ be the corresponding elliptic curve defined in the Introduction. Let

[TABLE]

be the smallest subset such that for every $E_{r^{\prime}}$ with $r^{\prime}\subseteq\tilde{L}_{r}$ we have $\pi_{r^{\prime}}(E_{r^{\prime}}[\infty])\subseteq\tilde{L}_{r}$.

Theorem 13**.**

[1*]**

Let $k$ be a number field. For every $a\in k\setminus\{0,\pm 1,\pm i\}$, and*

[TABLE]

the set

[TABLE]

is a field.

At first glance, it is rather surprising that such a simple and natural construction, inspired by comparisons of $x$-coordinates of torsion points of elliptic curves, produces a field. The conceptual reason for this is the rather peculiar structure of 4-torsion points of elliptic curves: translations by 2-torsion points yields, upon projection to ${\mathbb{P}}^{1}$, two standard commuting involutions on ${\mathbb{P}}^{1}(\bar{k})$, which allow to define addition and multiplication on $L_{a}$.

We may inquire about arithmetic and geometric properties of the fields $L_{a}$. For $a\in\bar{k}$ we let $k(a)\subseteq\bar{k}$ denote the smallest subfield containing $a$. We have:

• •

For every $a\in\bar{k}$, the field $L_{a}$ is a Galois extension of ${\mathbb{Q}}(a)$.

• •

For every $k$ of characteristic zero, $L_{a}$ contains ${\mathbb{Q}}^{ab}$, the maximal abelian extension of ${\mathbb{Q}}$.

• •

The field $L_{\zeta}$, where $\zeta$ is a primitive root of order $8$, is contained in any field $L_{a}$. Indeed, the corresponding elliptic curve $E$ has ramification subset

[TABLE]

which is projectively equivalent to $\{1,-1,i,-i\}\subset\pi(E[4])$. Since $\pi(E[4])$ projectively does not depend on the curve $E$, we obtain that $L_{\zeta}\subset L_{a}$, for all $a$. The same holds for $L_{a}$ where $E_{a}$ is isomorphic to $E_{3}$ (elliptic curve with an automorphism of order $3$).

• •

The field $L_{a}$ is contained in a field obtained as an iteration of Galois extensions with Galois groups either abelian or ${\rm PGL}_{2}(\mathbb{F}_{q})$, for various prime powers $q$. Is $L_{a}$ equal to such an extension? As soon as the absolute Galois group is not equal to a group of this type, e.g., for a number field $k$, we have

[TABLE]

• •

Let $a,a^{\prime}\in\bar{{\mathbb{Q}}}$ be algebraic numbers such that ${\mathbb{Q}}(a)={\mathbb{Q}}(a^{\prime})$. Then $L_{a}=L_{a^{\prime}}$. Varying $a\in\bar{{\mathbb{Q}}}$, we obtain a supply of interesting infinite extensions $L_{a}/{\mathbb{Q}}$.

The rest of this section is devoted to the proof of Theorem 13.

Proof.

Let $r_{0}:=\{0,\infty,1,-1\}$ and put $L:=\tilde{L}_{r_{0}}\setminus\{\infty\}$. Let

[TABLE]

be the elliptic curve with ramification in $r_{a}$. Since

[TABLE]

we have $L\subseteq L_{a}$, for all $a$. We first show that $L$ is a field.

Step 1. $L\backslash\{0\}$ is a multiplicative group. Indeed, for any $b\in L\backslash\{0\}$, we have

[TABLE]

and hence

[TABLE]

Since $b^{-1},-b^{-1}\in L$ we also have $\{0,1,-1,\infty\}\subset b\cdot L$. Thus $L\subseteq bL$. Similarly, $L\subseteq b^{-1}\cdot L$ or $b\cdot L\subseteq L$, which implies $L=bL$. Thus for any $a,b\in L$ we have $ab\in L$, and since the same holds for $ab^{-1}$, $b\neq 0$, we obtain $L\setminus\{0\}\subseteq\bar{k}^{\times}$.

Step 2. Let

[TABLE]

be the subgroup preserving $\tilde{L}$. It is nontrivial, since it contains $L\setminus\{0\}$ as a multiplicative subgroup, together with the involution $x\mapsto x^{-1}$. Consider

[TABLE]

It is an involution with $\gamma_{1}(\infty)=1,\gamma_{1}(0)=-1$ and hence $\gamma_{1}$ is coming from $r:=\{0,1,-1,\infty\}$. Thus it maps $L$ into $L$ and $\gamma_{1}\in{\rm Aut}_{L}$

Consider any pair of distinct elements $\{b,c\}\subset L$: it can be transformed into $\{0,1\}$ by an element from ${\rm Aut}_{L}$. If $b\neq 0,\infty$ then, dividing on $b$, we obtain $\{1,c/b\}$ and $\gamma_{1}(\{1,c/b\})=\{0,1\}$. If $b=0$ and $c\neq\infty$ then, dividing on $c$, we obtain $\{0,1\}$. If $b=0,c=\infty$ then $\gamma_{1}(\{0,\infty\})=\{-1,1\}$ and we reduce to the first case.

Step 3. $L$ is closed under addition. We show that $\gamma:x\mapsto x+1$ is contained in ${\rm Aut}_{L}$: by Step 2, there exists a $g\in{\rm Aut}_{L}$ which maps $\{-1,\infty\}$ to $\{0,\infty\}$ and hence $\{-1,0,\infty\}$ to $\{0,b,\infty\}$, for some $b\in L\setminus\{0\}$. Then $b^{-1}g\in{\rm Aut}_{L}$ maps $\{-1,0,\infty\}$ to $\{0,1,\infty\}$ and hence $b^{-1}g(x)=\gamma(x)=x+1$. Thus for any $a\in L$ we have $a+b=b(a/b+1)\in L$, which shows that $L$ is an abelian group.

Now let us turn to the general $L_{a}$.

Step 4. Note that $L\subset L_{a}$ and that $L_{a}$ is closed under taking square roots. Indeed for any $a\in L$ and $E_{r}$ with $r:=\{0,1,a,\infty\}$, we have $\sqrt{a}\in\pi_{r}(E_{r}[4])$ and hence $\sqrt{a}\in L_{a}$. Furthermore, for any $a,b\in L_{a}$ we have $\sqrt{ab}\in L_{a}$. Indeed, consider the curve $E_{r}$ with $r=\{0,a,b,\infty\}$. Then $\sqrt{ab}\in\pi(E_{r}[4])$, since the involution $z\to ab/z$ is contained in the subgroup ${\mathbb{Z}}/2\oplus{\mathbb{Z}}/2$ corresponding to the two-torsion on $E_{r}$, its invariant points are in $\pi_{r}(E_{r}[4])$. Iterating, we obtain that

[TABLE]

Step 5. For all $b\in L_{a},c\in L$ we have $\sqrt{b+c}\in L_{a}$. Indeed, for $c\in L$ we know that there is a solution $d\in L$ of the quadratic equation $d^{2}+d+c=0$. Consider the curve $E_{r}$ for $r:=\{\infty,b,d,d+1\}$. Then

[TABLE]

and hence $d\pm\sqrt{b-d}\in L_{a}$. Thus

[TABLE]

Step. 6. Let $P_{m}\in L[x]$ be a monic polynomial of degree $m$ and let $b\in L_{a}$. Then there is an $N(m)\in{\mathbb{N}}$ such that

[TABLE]

Indeed, we have

[TABLE]

The statement holds for $m=1$ by Step 4. Assume that it holds for $m-1$. Then $c_{m-1}+b(c_{m-2}+\cdots)=d^{4^{N(m-1)}}$ for some $d\in L_{a}$. We can then write

[TABLE]

by taking $t=\sqrt[4^{N(m-1)}]{b}$ and $u_{m}=\sqrt[4^{N(m-1)}]{c_{m}}$ we obtain

[TABLE]

where $t\in L_{a},u_{m}\in L$ and $\zeta^{i}$ runs through the roots of unity of order $4^{N(m-1)}$.

By Steps 4 and 5, we obtain that $4^{m}4^{N(m-1)}$-th root of $P_{m}(b)$ is contained in $L_{a}$, thus the result holds for $N(m)=4^{N(m-1)}$

Step 7. Let $b\in L_{a}$ be any algebraic element over $L$. Then the field $L(b)$ is a finite extension of $L$ and there is an $n\in{\mathbb{N}}$ such that any $x\in L(b)$ can be represented as a monic polynomial of $b$ with coefficients in $L$ of degree $\leq n$. For such $n$ we define a power $4^{N}$ such $\sqrt[4N]{x}\in L_{a}$, but then any element in $L(b)$ is in $L_{a}$.

∎

Remark 14*.*

In the proof we have only used points in $\pi(E[4])$. Therefore, for any subset $D\subset{\mathbb{N}}$ containing $4$ we can define $L_{a,D}$, as the smallest subset containing all $\pi(E[n])$ for all $n\in D$ and all elliptic curves obtained as double covers with ramification in $L_{a,D}$. It will also be a field.

For example, if $D=\{3,4\}$ then $L_{a,D}$ is exactly the closure of $L_{a}$ under abelian degree $2$ and $3$ extensions, since ${\rm PGL}_{2}(\mathbb{F}_{2})=\mathfrak{S}_{3}$ and ${\rm PGL}_{2}(\mathbb{F}_{3})=\mathfrak{S}_{4}$ and both groups are solvable with abelian quotients of exponent $3,2$.

On $({\rm Sym}^{4}({\mathbb{P}}^{1}(\bar{\mathbb{Q}}))\setminus\Delta)/{\rm PGL}_{2}(\bar{\mathbb{Q}})$ we can define a directed graph structure $DGS$, postulating that

[TABLE]

if there is an elliptic curve $E^{\prime}$ isogeneous to $E_{r_{z}}$ such that $r_{w}$ is projectively equivalent to a subset in $\pi(E^{\prime}[\infty])$. Any path in the graph is equivalent to a path contained in $({\rm Sym}^{4}({\mathbb{P}}^{1}(L(E)))\setminus\Delta)/{\rm PGL}_{2}(\bar{\mathbb{Q}})$, for some $E$. The graph contains cycles, periodic orbits, and preperiodic orbits, i.e., paths which at some moment end in periodic orbits.

Question 15*.*

Consider the field $L_{0}=L_{r_{0}}$ for $r_{0}=\{0,1,-1,\infty\}$. Does

[TABLE]

consist of one cycle in $DGS$? Note that any path beginning from $r_{0}$ extends to a cycle (in many different ways) since $r_{0}$ is ${\rm PGL}_{2}$-equivalent to a four-tuple of points of order $4$ on any elliptic curve.

Remark 16*.*

In Step 7, we have used algebraicity of $L_{a}/L$, and we do not know how to extend the proof to geometric fields. What are the properties of $L_{a}$ in geometric situations, when $a$ is transcendental over $k$?

We have seen in the proof that the field $L_{a}$ is closed under extensions of degree 2. We also have:

Lemma 17**.**

For any $b\in L_{a}$, we have $\sqrt[3]{b}\in L_{a}$.

Proof.

Consider a curve $E_{r}$ with $r:=\{b,\sqrt{b},-\sqrt{b},\infty\}$. Its 3-division polynomial takes the form:

[TABLE]

We can represent it as a product: $3\prod_{i=1}^{4}(x-x_{i})$, where the set $\{x_{i}\}\subset L_{a}$ is equal $\pi(E_{r}[3])$. The corresponding cubic resolvent

[TABLE]

where $(i,j),(k,l)$ is any splitting into pairs of indices among $1,2,3,4$. In terms of $b$, we have

[TABLE]

Since the set $\{x_{i}\}$ is projectively equivalent to $\{0,1,\zeta_{3},\zeta_{3}^{2}\}$, we can see that the cubic polynomial above has the form $C(x^{3}+B)$, for some constants $C,B$. It can be checked that

[TABLE]

After a projective map in ${\rm PGL}_{2}(L_{a})$ we can transform the the elements $x_{i}x_{j}+x_{k}x_{l}$ into $-\sqrt[3]{(b-1)^{2}}$. Hence $-\sqrt[3]{(b-1)^{2}}\in L_{a}$, for any $b\in L_{a}$; since $L_{a}$ is a field closed under $2$-extensions we obtain the claim. ∎

This raises a natural

Question 18*.*

Is $L_{a}$ is closed under taking roots of arbitrary degree?

If we add $\mathbb{G}_{m}$ to the set of allowed elliptic curves then the answer is affirmative. However, there may exist a purely elliptic substitute for obtaining roots of prescribed order.

Corollary 19**.**

If the $j(E)\in L_{a}$ then any set $\{b,-b,b^{-1},-b^{-1}\}$ with $\mu_{4}((b,-b,b^{-1},-b^{-1}))=j(E)$ is contained in $L_{a}$. Note that such $b$ are solutions of a cubic equation. Thus $L_{a}$ depends only on the curve $E$ and we will write $L(E)$.

It is also easy to see that $L(E)=L(E^{\prime})$ if $E$ and $E^{\prime}$ are isogenous.

5. Intersections

In this section we present further results concerning intersections

[TABLE]

for different elliptic curves $E_{1},E_{2}$ and provide evidence for the Effective Finiteness Conjecture 2.

Proposition 20**.**

Assume that

[TABLE]

and that

[TABLE]

Then $r_{1}=r_{2}$ and $E_{1}=E_{2}$.

Proof.

By our assumption (2), $E_{i}$ are given by the equation

[TABLE]

With $a_{i}$ defined by

[TABLE]

we have

[TABLE]

We assume that $\pi_{i}(e_{i})=a_{i}$. In this case, points $\pi_{i}(E_{i}[3])\subset\bar{{\mathbb{Q}}}\subset{\mathbb{P}}^{1}$ are the roots of

[TABLE]

or, equivalently,

[TABLE]

If $x,y\in\pi_{a_{1}}(E_{a_{1}}[3])\cap\pi_{a_{2}}(E_{a_{2}}[3])$, where $x\neq y$ and $a_{1}\neq a_{2}$, then $a_{1}$ and $a_{2}$ are the roots of $2x^{3}a^{2}+(x^{4}-1)a-2x$ and of $2y^{3}a^{2}+(y^{4}-1)a-2y$, that means that their coefficients are proportional

[TABLE]

Then, on the one hand, $x^{3}/y^{3}=x/y$ implies $x^{2}=y^{2}$, and hence $x=-y$, by our assumption that $x\neq y$. On the other hand,

[TABLE]

a contradiction. ∎

Given any $x\in\bar{{\mathbb{Q}}}$ we obtain $a_{i}=a_{i}(x)$, $i=1,2$, which satisfy (3). Then the resulting elliptic curves $E_{i}$ satisfy (2) and we have

[TABLE]

unless

[TABLE]

Moreover,

[TABLE]

where $6$ is the number of images of common points of order $4$ (from Equation 2) and $4$ stands for the size of $({\mathbb{Z}}/2\oplus{\mathbb{Z}}/2)$-orbit of a point in ${\mathbb{P}}^{1}$. However, it may happen that the inequality in (4) is strict.

Example 21*.*

Consider the polynomial $f_{5}(x,a)$ defined in [2, Theorem 18]). Its roots are exactly $\pi_{a}(E_{a}[5])$. It has degree $12$ with respect to $x$ and $6$ with respect to $a$. The polynomial $f_{3}(x,a)$ has degree $2$ with respect to $a$ and generically has exactly two solutions $a_{1}(z),a_{2}(z)$, for any given $z$. We want also $f_{5}(v,a_{i}(z))=0$ for some $v$ and $z$. This is equivalent to $f_{5}(v,a)$ being divisible by $f_{3}(z,a)$, as polynomials in $a$. Writing division with remainder

[TABLE]

for some explicit polynomials $C$, and $C^{\prime}$, which have to vanish. This condition is gives an explicit polynomial in $u$, which is divisible by a high power of $u$ and $(u-1)$. Excluding the trivial solutions $u=0,1$, and substituting $t=u^{4}$ we obtain the equation

[TABLE]

Since $t\neq 0$, we have

[TABLE]

Computing the discriminant of this cubic polynomial, we find that it has no multiple roots. Its solutions give rise to pairs $u,v$ such that for $a_{1}:=a_{1}(u),a_{2}:=a_{2}(u)$ we have

[TABLE]

and hence

[TABLE]

The symmetry of the above equation reduced the problem to a cubic equation with coefficients in ${\mathbb{Q}}$, followed by a quadratic equation. The roots can be expressed in closed form and hence we get explicit description for the 24 roots $u$.

The same scheme can be applied to points of higher order. Indeed we have a polynomial $f_{n}(u,x)=0$ which has increasing degree with respect to $u$, and the existence of a pair $u,v$ such that $f_{n}(v,x)=0$ is divisible by $f_{3}(u,x)$ depend on the divisibilty of $f_{n}(v,x)$ by $f_{3}(u,x)$. Using long division we obtain two polynomials $C_{0,n}(u,v)$ and $C_{1,n}(u,v)$ so that their common zeroes $(u,v)$ correspond to pairs $(u,v)$ with $f_{3}(u,x)=0$ and $f_{n}(v,x)=0$ simulaneously.

Example 22*.*

Applying this scheme to points of order $3$ and $7$ (or 3 and 11, 3 and 13, 3 and 17) we obtain that the corresponding resultant has roots of multiplicity three which implies the existence of three points $v$ for a given $u$ with $f_{3}(u,x)=0$ and $f_{7}(v,x)=0$ and hence

[TABLE]

Since we have every reason to expect polynomials $C_{0,n}(u,v)$ and $C_{1,n}(u,v)$ to have increasing number of intersection points with the growth of $n$ we are led to the following conjecture:

Conjecture 23**.**

There is an infinite dense subset of points $a\in{\mathbb{P}}^{1}$ such that

[TABLE]

with

[TABLE]

Note that in all such cases the fields $L_{a}=L_{a_{2}}$.

6. General Weierstrass families

The family of elliptic curves considered in Section 5 is the most promising for obtaining large intersections of torsion points. In this section, we consider other families where the intersections tend to be smaller, following [2].

We consider elliptic curves $E_{a}$ with the same

[TABLE]

These are given by their Weierstrass form

[TABLE]

Using formulas in, e.g., [3, III, Section 2], we write down (modified) division polynomials $f_{n,a}$, whose zeroes are exactly $\pi_{a}(E_{a}[n])$:

[TABLE]

where $d(n)$ and the coefficients $c_{r,s,t}(n)$ can be expressed via totient functions $J_{k}(n)$, with $d(n)=J_{2}(n)/2$, if $n>2$, and $d(2)=3$ (see [2]).

Lemma 24**.**

Let $E_{1},E_{2}$ be elliptic curves in generalized Weierstrass form (5) such that, for some $n>1$ we have

[TABLE]

Then $E_{1}\simeq E_{2}$.

Proof.

The statement is trivial for $n=2$. For $n>2$, we have $d(n)\geq 4$, the comparison of division polynomials implies that the terms

[TABLE]

must be equal. For

[TABLE]

we find equality of coefficients $a_{i}$ for both curves. ∎

Often, already the existence of nontrivial intersections

[TABLE]

leads to the isomorphism of curves $E_{1},E_{2}$. For example, if both curves are defined over a number field $k$ and the action of the absolute Galois group $G_{k}$ on $\pi_{1}(E_{1}[n])$ and $\pi(E_{2}[n])$ is transitive then (6) implies that $E_{1}\simeq E_{2}$. For many, but not all, $n\in{\mathbb{N}},$ the equality of totient functions $J_{2}(n)=J_{2}(m)$, for some $m\in{\mathbb{N}}$, implies $n=m$.

Example 25*.*

There exist many tuples $(m,n)$ for which

[TABLE]

For example,

[TABLE]

We also have

[TABLE]

On the other hand, we have

[TABLE]

These results indicate a relation of our question to Serre’s conjecture. He considered the action of the Galois group on torsion points of an elliptic curve $E$ defined over a number field $k$. If $E$ does not have complex multiplication, then the image of the absolute Galois group $G_{k}$ is an open subgroup of ${\rm GL}_{2}(\hat{\mathbb{Z}})$, i.e., of finite index.

Conjecture 26** (Serre).**

For any number field $k$ there exists a constant $c=c(k)$ such that for every non-CM elliptic curve $E$ over $k$ the index of the image of the Galois group $G_{k}$ in ${\rm GL}_{2}(\hat{\mathbb{Z}})$ is smaller than $c$.

In particular, for $k={\mathbb{Q}}$ he conjectured that for primes $\ell\geq 37$ the image of $G_{k}$ surjects onto ${\rm PGL}_{2}({\mathbb{Z}}_{\ell})$. Thus, modulo Serre’s conjecture, our conjecture holds for curves defined over ${\mathbb{Q}}$.

Proposition 27**.**

Assume that

[TABLE]

Then $k(E[n])$ contains ${\mathbb{Q}}(\zeta_{d})$, where $d=\mathrm{lcm}(m,n)$, the least common multiple of $m,n$.

Proof.

By Serre, we have

[TABLE]

as subfields of index at most $2$. ∎

Corollary 28**.**

Assume that $k$ does not contain roots of $1$ of order divisible by $n,m$. Then $k(E[n]),k(E(m))$ contain a cyclotomic subfield of index at most $2$.

This provides a strong restriction on intersections of images of torsion points for elliptic curves over ${\mathbb{Q}}$, or over more general number fields $k$ with this property. This yields a restriction on fields $k(E[n])$, since $(n,m)>4$, for all $(n,m)$ with $J_{2}(n)=J_{2}(m)$.